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I've seen similar questions here, and I wish there was a narrower tag than the broad soft-question and career-development.

I am currently a freshman/sophomore-to-be at a top ten math program in US. My major GPA is around 3.5-3.6, and so is my overall GPA. Due to fast progress in my courses (skipped all lower divs) I will be ready to graduate in a year or year and a half. The question is whether I should do that.

I do not see myself outside of academia, and dead-set on pursuing PhD in math. With my GPA far from being stellar, I was going to take more grad courses to improve the situation. The college is quite pricey, with me being an out-of-state, and so I am not really sure whether I should just graduate and take those courses back home (or apply for masters).

Recap: is it better to graduate early, with an average GPA and no hooks (e.g., research, high Putnam grade, no grad courses), apply for Masters program and save money, or graduate later and improve my record as an undergrad?

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How good is your relationship with your professors? (I mean: how good will their recommendation letters be?) –  Qiaochu Yuan Aug 1 '12 at 2:37
    
Only 2 courses I've taken were taught by professors. I am really trying to seek attention of one very influential number theorist, and I do not think it's worked so far. –  user22835 Aug 1 '12 at 2:47
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@mixedmath Too much of a giveaway. I wouldn't want a professor to think that I'm after him just because of the recommendation letters. –  user22835 Aug 1 '12 at 4:08
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Have you looked into establishing residency in the state? I think the difficulty of this task varies substantially from one state to another, but it may be worth looking into. –  Dan C Aug 2 '12 at 1:46
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It is extremely difficult to establish residency in California. –  user22835 Aug 4 '12 at 22:33
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4 Answers

Your answer will depend partly on what you want to do in academia. If you want to teach, but don't really want to emphasize research much, you might do fine to graduate now. However, if your goal is to become a professor at a research intensive school, then you really should go to the strongest grad school you can get into. (Based on your description, I strongly suspect that if you bust your butt for another year or two, in particular working to earn one or two strong letters of rec, you could get into a better grad school than you can currently.)

Yes, I know there are considerations about who you will work with, perhaps geography, potential two-body problems, etc. So, why's it so important to go to the best school you can? Again and again I see that in academia (as everywhere) networking is crucial. Generalizing and stereotyping a bit: the best schools have the best researchers, who know the other best researchers, who have the biggest grants, which fund the nicest postdocs, etc. If you want to thrive as a researcher, you will do well to get into that network. (To a large extent, it's a rich get richer system.) As an undergrad or early grad student, one way you can get into that network is to work with a professor who is a central part of it, and is willing to weave you in. And your chances of working with said professor typically go up with the reputation of the school.

Now a personal digression. Through high school and undergrad I was in a hurry to get to the next level as soon as possible. I skipped 7th grade, finished undergrad in 3 years, and started grad school at the age of 20. I even turned down a year abroad in the Budapest Semester in Math, because I was worried I'd miss out if I waited too long to get to grad school. The sad truth is that I wasn't ready. Maybe you would be; I've never met you, so I can't say.

Eventually (after 8 years), I finished a PhD, and am fairly happy with where my career is headed. However, I don't regret that time in grad school at all. I learned a lot of useful stuff. In fact, I think it's because I took my time in grad school that my career has gone as well as it has. One interesting feature of academia is that you're typically judged by your productivity relative to the time since you earned your PhD (rather than your age). As a result, I encourage you to take your time and learn as much as you can. You'll never again have as much free time as you do now.

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I want to add a couple of remarks to Dan C's excellent answer.

No, you should not graduate early, unless your finances force it. Except for cost, there is no advantage to finishing early. Admissions committees will compare you to other undergraduate applicants, not other applicants your age; age discrimination is illegal. They will not care about your grades, as long as they're good; everyone applying to those departments has fantastic grades. I suspect almost all applicants to top math departments skipped their low-level math classes. I assume you'll ace the math GRE, but then so will everyone else.

What sets the successful applicants to top PhD programs apart is strong evidence of research potential. To gain admission to the top departments you must have strong letters from faculty that directly praise your potential for mathematical research in specific, personal, and credible detail. The only way to get those letters is to work directly with faculty outside the classroom. Fortunately, because you skipped your freshman classes, you have room in your schedule to do that.

You mention "out of state", so I'm guessing you're at a big state school (like mine); your classes so far probably had hundreds of students, and many were taught by graduate students or adjuncts. You have to break out of that; you must get to know some faculty. And because you're at a big state school, this is going to require considerable initiative on your part.

Taking graduate classes is a good way to meet faculty, but it's just a start. Do not just sit quietly in class and get an A. Meet with your instructor early in the semester, explain you ambition for academia, and ask about opportunities for research and/or independent study. Be prepared to explain what kind of mathematics you're best at, and what kind of math inspires you. The first prof you talk to is not likely to be a good match for your specific interests; ask them for suggestions of other faculty to talk to. Repeat ad infinitum.

There are many other reasons not to graduate early, which are less professional, but no less important. Give yourself some time to grow up.

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Too long to be a comment, and originally written for Math.SE -

Firstly, if you're set on a math PhD, then you will probably never apply for a Master's program. Most math PhDs apply straight to PhD, and these are generally funded (this is all under the assumption that you stay in the US).

The typical accepted candidate to a good math PhD program has a good background in the following:

  1. GPA
  2. Research
  3. Math knowledge
  4. GRE/Math GRE
  5. Recommendations

One doesn't need to be perfect at everything, and everyone has their strengths and weaknesses. The exception to this rule is that you must have great recs - there is a certain recommendation inflation right now, and it seems to me that recommendations are judged just as harshly by what is not said. I say this only because without grad courses/research/high gpa, it might be challenging for one of your professors to speak highly on your behalf. Or maybe not - it's case by case, right?

I cannot speak as to how strong your exact application would be, as I don't know the specifics. Your best feedback would probably come from a mentor or advisor from your department, or one of the professors whom you would ask for a recommendation.

Without knowing specifics, I might also ask: what is the rush for? (rhetorically)

As a final note, I should mention that it might be possible, depending on your school's policy, to apply for grad programs and decide to actually graduate only if you get accepted/have positive feedback. But this is not ideal, as it's sort of a punt. Were you to not get accepted, you wouldn't have set up summer plans and your last year would be somewhat hodge-podge. Yet these are the exact things that would improve your application for the next year.

Food for thought.

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The OP mentions the cost of college as a possible reason to graduate early. –  Qiaochu Yuan Aug 1 '12 at 3:24
    
Thanks, Qiaochu. Yes, the cost hits me pretty bad, being a recent immigrant of more than humble backgrounds, and the whole package associated with it (e.g.,first family member to have his diploma recognized in US) –  user22835 Aug 1 '12 at 4:25
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I would reorder your list as 1. Research 2. Recommendations. 3. Everything else. –  JeffE Aug 1 '12 at 9:27
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To complement other answers: the "research" issue is volatile. Ill-informed, childish "research" is not a plus in applications to elite grad programs. I think it might be more apt to be able to give evidence of _getting_in_sync_with_ some contemporary serious research, even if one isn't yet able to make one's own contribution. Better to be an apprentice at something serious, than journeyman at something of dubious interest to professionals.

And, the same thing said in a different context: coursework per se is nice, but is (almost entirely) miles away from live mathematics. Thus, the point is not to "bluff" "research", but to get beyond the sterile, formal classroom/textbook mathematics. Live discussions with faculty and presence in seminars are substantive steps in the right direction, without the too-facile pretense that one is doing a big research project in 10 weeks in the summer, based on scant prior information. :)

Being able to have people speak on your behalf, that you have ably moved beyond "school math", and that that's what you are interest in, and have talent for (never mind classrooms) is what will get you into an elite program.

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